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Numbers following gaps in the sequence of base-3 numbers that don't contain 0.
1

%I #36 Oct 13 2023 14:20:17

%S 1,4,7,13,16,22,25,40,43,49,52,67,70,76,79,121,124,130,133,148,151,

%T 157,160,202,205,211,214,229,232,238,241,364,367,373,376,391,394,400,

%U 403,445,448,454,457,472,475,481,484,607,610,616,619,634,637,643,646,688,691,697,700,715,718,724,727,1093,1096,1102,1105,1120

%N Numbers following gaps in the sequence of base-3 numbers that don't contain 0.

%C Partial sums for the convergent modified harmonic series in base 3 excluding 0 = Sum of 1/a(n) + 1/(a(n) + 1) = Sum of (2*a(n) + 1)/(a(n)*(a(n) + 1)).

%H Robert Baillie, <a href="http://dx.doi.org/10.2307/2321096">Sums of Reciprocals of Integers Missing a Given Digit</a>, American Mathematical Monthly (Washington, DC: Mathematical Association of America) 86 (5): 372-374, May 1979, doi:10.2307/2321096. ISSN 0002-9890. JSTOR 2321096.

%F a(n) = A032924(2n - 1).

%e Pattern of numbers of skipped terms (numbers in base 3 with at least one zero) is 1 (3 = 10_3), 1 (6 = 20_3), 3+1 (9 = 100_3, 10 = 101_3, 11 = 102_3, 12 = 110_3), 1, 3+1, 1, 9+3+1, 1, 3+1, 1, 9+3+1, 1, 3+1, 1, 27+9+3+1, ...

%o (PARI) lista(nn)=prec0 = 1; for(n=1, nn, if (vecmin(digits(n, 3)), if (prec0, print1(n,, ", ")); prec0 = 0, prec0 = 1);); \\ _Michel Marcus_, Aug 03 2015

%o (Python)

%o def A259566(n): return int(bin(m:=n)[3:],3)*3 + (3**m.bit_length()-1>>1) if n>1 else 1 # _Chai Wah Wu_, Oct 13 2023

%Y Cf. A032924.

%Y Subset of A016777 (congruent to 1 mod 3).

%Y Each term is one more than each number that follows a gap in A081605.

%K nonn,base,less

%O 1,2

%A _Sean Oneil_, Jun 30 2015